218
7 Bounded Quantum Systems
where J (u, v) = (d + aξ(u)) is the Jacobian of the coordinate transformation. In
terms of the coordinates (u, v), the billiard now has the shape of a rectangle with
straight sides of length 2π along the u-direction and length 1 along the v-direction.
For the case when all four walls of the billiard are hard, we can expand the
eigenfunctions in terms of an orthogonal basis, φ l,m (u, v), so that
n (u, v) =
∞
l=1
∞
m=1
B
(n)
l,m φ l,m (u, v),
(7.34)
with
φ l,m (u, v) =
2
π
1
√
J
sin(lu)sin(mπ v).
(7.35)
The orthonormal basis set, φ l,m (u, v), can be used to construct a Hamiltonian
matrix, H lml m , and an eigenvalue equation (from Eq. (7.30)) such that
∞
l=1
∞
m=1
H lml m B
(n)
l m = E n B
(n)
lm .
(7.36)
The eigenvalues, E n , and coefficients, B
(n)
lm , can be determined from Eq. (7.36).
Once the coefficients B
(n)
lm are known, the eigenvectors n (u, v) and ψ n (x, y) can
be constructed. The nearest neighbor eigenvalue spacing distribution for a ripple
billiard in a parameter regime where the classical dynamics is fully chaotic has
been obtained in Li et al. (2002). After contributions from bouncing ball orbits are
removed, the nearest neighbor spacing statistics agree with the GOE prediction with
a 99% confidence level.
Once the energy eigenstates, ψ n (x, y), have been obtained, they can be used to
construct a quantum Poincaré surface of section using Husimi distributions. Husimi
distributions give a coarse-grained view of the distribution of probability in phase
space for a given quantum state. A quantum Poincaré surface of section can be
constructed if we use the fact that there is an approximate separation of variables
near the billiard walls (Crespi et al. 1993; Luna-Acosta et al. 1996). Near y = 0, the
wave function ψ n (x, y) can be separated to first order,
ψ n (x, y) | y≈0 = 0 +
∂
∂y
ψ n (x, y) | y=0 y + O(y)
2
+ . . . .
(7.37)
We can define a new wave function,
S n (x)≡
∂
∂y
ψ n (x, y) | y=0 ,
(7.38)
7 Bounded Quantum Systems
where J (u, v) = (d + aξ(u)) is the Jacobian of the coordinate transformation. In
terms of the coordinates (u, v), the billiard now has the shape of a rectangle with
straight sides of length 2π along the u-direction and length 1 along the v-direction.
For the case when all four walls of the billiard are hard, we can expand the
eigenfunctions in terms of an orthogonal basis, φ l,m (u, v), so that
n (u, v) =
∞
l=1
∞
m=1
B
(n)
l,m φ l,m (u, v),
(7.34)
with
φ l,m (u, v) =
2
π
1
√
J
sin(lu)sin(mπ v).
(7.35)
The orthonormal basis set, φ l,m (u, v), can be used to construct a Hamiltonian
matrix, H lml m , and an eigenvalue equation (from Eq. (7.30)) such that
∞
l=1
∞
m=1
H lml m B
(n)
l m = E n B
(n)
lm .
(7.36)
The eigenvalues, E n , and coefficients, B
(n)
lm , can be determined from Eq. (7.36).
Once the coefficients B
(n)
lm are known, the eigenvectors n (u, v) and ψ n (x, y) can
be constructed. The nearest neighbor eigenvalue spacing distribution for a ripple
billiard in a parameter regime where the classical dynamics is fully chaotic has
been obtained in Li et al. (2002). After contributions from bouncing ball orbits are
removed, the nearest neighbor spacing statistics agree with the GOE prediction with
a 99% confidence level.
Once the energy eigenstates, ψ n (x, y), have been obtained, they can be used to
construct a quantum Poincaré surface of section using Husimi distributions. Husimi
distributions give a coarse-grained view of the distribution of probability in phase
space for a given quantum state. A quantum Poincaré surface of section can be
constructed if we use the fact that there is an approximate separation of variables
near the billiard walls (Crespi et al. 1993; Luna-Acosta et al. 1996). Near y = 0, the
wave function ψ n (x, y) can be separated to first order,
ψ n (x, y) | y≈0 = 0 +
∂
∂y
ψ n (x, y) | y=0 y + O(y)
2
+ . . . .
(7.37)
We can define a new wave function,
S n (x)≡
∂
∂y
ψ n (x, y) | y=0 ,
(7.38)